The American Swedish Institute in Mpls is free one day a month. Check their website, but it's something like the first Thursday of the month. The Walker Art Museum--this would be for you and your husband, not the kids--is free every Thursday from 5:00 p.m. to close (8:00 or 9:00 p.m.). The Art Institute is *always* free for all ages. (Some exhibits require an admission fee, but they are few and far between.)
I echo the recommendation to make good use of the library system. Merriam Park library has a *great* kids' area. We are always checking out videos and books at the library, and we go to storytime.
I don't know where in St. Paul you are, but there are two West 7th Community Centers, and they have different things going on in the mornings and early afternoons during the weekdays. One day is a folk singer, another is a science type project, another is art. They have a donation box, but it is free.